We present the results of an intermediate-resolution (1.5 A) spectroscopic study of 17 x-ray luminous narrow emission-line galaxies previously identified in the Cambridge-Cambridge ROSAT Serendipity Survey and the Einstein Extended Medium Sensitivity Survey. Emission-line ratios reveal that the sample is composed of ten Seyfert and seven starburst galaxies. Measured linewidths for the narrow H alpha emission lines lie in the range 170 - 460 km s(exp -1). Five of the objects show clear evidence for asymmetry in the (OIII) lambda 5007 emission-line profile. Broad H alpha emission is detected in six of the Seyfert galaxies, which range in type from Seyfert 1.5 to 2. Broad H beta emission is only detected in one Seyfert galaxy. The mean full width at half maximum for the broad lines in the Seyfert galaxies is FWHM = 3900 +/- 1750 km s(exp -1). Broad (FWHM = 2200 +/- 600 km s(exp -1) H alpha emission is also detected in three of the starburst galaxies, which could originate from stellar winds or supernovae remnants. The mean Balmer decrement for the sample is H alpha / H beta = 3, consistent with little or no reddening for the bulk of the sample. There is no evidence for any trend with x-ray luminosity in the ratio of starburst galaxies to Seyfert galaxies. Based on our previous observations, it is therefore likely that both classes of object comprise approximately 10 percent of the 2 keV x-ray background.